PERPUSTAKAAN KU; TTHO

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Saya

PSZ 19: 16 (Pind. 1 97)

UNIVERSITI TEKNOLOGI l\lALA YSIA

BORANG PENGESAHAN STATUS TESIS·

JUDUL: A COMPARISON OF NUMERICAL METHODS FOR SOLVING THE BRATU AND BRATU-TYPE PROBLEMS

SESI PENGA.JIAN: 2004/2005

RUHAILA BINn MD. KASMANI (IIURUF BESAR)

mcngaku mcmbcnarkan tcsis (PSM-I Sarjana 1 GektBr-f<filsafah)* ini disimpan di Pcrpustakaan Univcrsiti Tcknologi Malaysia dcngan syarat-syarat kcgunaan scpcrti bcrikut:

I. Tcsis adalah hakmilik Univcrsiti Tcknologi Malaysia. 2. Pcrpustakaan Univcrsiti Tcknologi Malaysia dibcnarkan membuat salinan untuk

tujuan pengajian sahaja. 3. Pcrpustakaan dibenarkan mcmbuat salinan tesis ini scbagai bah an pcrtukaran antara

institusi pcngajian tinggi. 4. ** Sila tandakan (,j)

D SULIT

D TERHAD

CD TIDAK TERHAD

(TANDATANGAN PENULlS)

Alamat Tctap: 36, .lIn. Tcratai 4, Taman Bakri Indah, 84200 Bakri, 1\1uar, .Johor.

Tarikh: 16 Mac 2005

CA T A TAN * Potong yang tidak berkenaan .

(Mcngandungi maklumat yang bcrdarjah kcsclamatan atau kcpcntingan Malaysia scpcrti yang tcrmaktub di dalam AKTA RAHSIA RASMI 1972)

(Mcngandungi maklumat TERHAD yang tclah ditcntukan olch organisasifbadan di mana pcnyclidikan dijalankan)

Disahkan olch

(TANDATANGAN I'ENYELlA)

":\1 DR. ALI AIm. RAII:\L\:\ Nama Pcnyclia

Tarikh: 16 i\lac 2005

•• Jib tesis ini SULlT atau TERIIAD. sib lampirbn surat dJfipada plhak t>crkulSa./or"an''3si berkcnaan dengan mcnyatabn sekali scbab dan tcmpoh tcsis ini pcrlu dikclasbn sch"p' SULlT atau TERHAD .

• Tcsis dimaksudbn sebagai tcsis bagi Ijazah Doktor ralS3fah dan Sarpna SCc.1ra !"cmclidlbn. mau discrtai Dagi pcngajian sccara kcqa kursus dan penyclidibn. atau Laroran Pcold. Sorl"n" 1\luda (I'S1\1)

"I hereby declare that I have read this thesis and in my

opinion this thesis is sufficient in terms of scope and quality for the

award of the degree of Ivlaster of Science (Mathematics)"

Signat.ure '{~l ,oJ ~ 2_~! <-c· <'~,

C· .. · .. · .. ·(j'··············· .. ·····~········· .. ··

Name of Sup~rvisor Assoc. Prof. Dr. Ali Abd.Rahman

Date 16 Ivlarch 2005

A COIvIPARISOIi OF NUlvIERICAL IvIETHODS FOR SOLVING

THE BRATU AND BRATU-TYPE PROBLEMS

RUHAILA IvID.KASMANI

A dissertation submitted in partial fulfilment of the

requirements for the award of the degree of

Master of Science UVlathematics)

Faculty of Science

Universiti Telmologi Malaysia

t-./IARCH 2005

11

I declare that this thesis entitled "A Comparison of Numerical

Methods for Solving the Bmtu and Bmtu-Type Problems" is the

result of my own research except as cited in the references. The

thesis has not been accepted for any degree and is not concurrently

submitted in candidature of any degree.

Signature

Name Ruhaila Md.Kasmani

Date 16 March 2005

III

Tp my beloved family and friends.

iv

ACKNOWLEDGEMENTS

In the name of Allah, The Most Gracious and The Ivlost ivIerciful. First

of all, thanks to Almight~r Allah for graciously bestowing me the perseverance to

tmdertake this study.

I would like to express my heartfelt gratitude to my supervisor, Associate

Professor Dr. Ali Abd.Rahman for his valuable advise and great encouragement

as well as for his excellent guidance and assistance for this dissertation.

A special thanks and a deepest appreciation to my beloved sister, Rafiziana

IVld.Kasmani for her cooperation and undivided support in completion of this

work. A warmest gratitude dedicates to my parents for their patience, prayers,

and encouragement thro~lghout my studies here in UTlvI .

Lastly, special thanks are extended to my friends for their cooperation and

assistance.

v

ABSTRACT

The Bratu problem ul/(x) + /\eu(x) = 0 with u(O) = u(l) = 0 has two

exact solutions for values of 0 < A < Ac, no solutions if A > Ac while unique

solution is obtained when A = Ac where Ac = 3.513830719 is the critical

value. The First Bratu-Type problem corresponds A = _7[2 while the Second

Bratu-Type problem is ul/(x) + 7[2e-u(x) = o. The exact solution of the First

Bratu-Type problem blows up at x = 0.5 while the Second Bratu-Type problem

is continuous. The present work seeks to compare various numerical methods

for solving the Bratu and Bratu-Type problems. The numerical methods are the

standard Adomian decomposition method, the modified Adomian decomposition

method, the shooting method and the finite difference method. These methods

are implemented using Maple. Convergence is achieved by applying the four

methods when 0 < A ::; 2, however the shooting method is the most effective

method as it gives the smallest maximum absolute error. ·When A = Ac, none of

these methods give the convergence solutions. Due to the nature of the solution

of the First Bratu-Type problem, only the shooting method and the modified

Adomian decomposition method can give the convergence values to the exact

solution. The finite difference method is proved to be the most effective method

for the Second Bratu-Type problem compared to other methods.

Keywords: Bratu problem, Bratu-Type problems, standard Adomian decomposi­

tion method, modified Adomian decomposition method, shooting method, finite

difference method.

vi

ABSTRAK

Masalah Bratu UIl(X)+AeU(X) = 0 dengan syarat sempadan v.(O) = u(l) = 0

mempunyai dua penyelesaian bagi 0 < A < Ae , tiada penyelesaian jika /\ > /\

dan penyelesaian unik jika A = Ae di mana Ae = 3.513830719 adalah merupakan

nilai genting. rvIasalah Jenis-Bratu Pertama mengambil nilai A = _1[2 manakala

masalah .Ienis-Bratu Kedua adalah u"(x) + 1[2e-u (x) = O. Penyelesaian sebe­

nar bagi masalah Jenis-Bratu Pertama meningkat secara mendadak pada titik

x = 0.5 manakala penyelesaian sebenar masalah Jenis-Bratu Kedua adalah selan­

jar. Disertasi ini melaporkan mengenai perbandingan di antara beberapa kaedah

berangka bagi menyelesaikan masalah Bratu dan Jenis-Bratu. Perbandingan ini

melibatkan penggunaan empat kaedah iaitu kaedah penghuraian Adomian asal,

kaedah penghuraian Adomian terubahsuai, kaedah penembakan dan kaedah pem­

beza terhingga. Setiap penyelesaian berangka dilaksanakan dengan menggunakan

Maple. Bagi kes 0 < A ::::; 2, keempat-empat kaedah tersebut telah memberikan

penyelesaian berangka yang menumpu. Didapati kaedah penembakan merupakan

kaedah yang paling efektif. Hanya kaedah penembakan dan kaedah penghuraian

Adomian terubahsuai telah menunjukkan penyelesaian menumpu bagi masalah

Jenis-Bratu Pertama. Kaedah pembeza terhingga merupakan kaedah yang pal­

ing efektif bagi mendapatkan penyelesaian berangka untuk masalah Jenis-Bratu

Kedua berbanding kaedah yang lain.

Kata kunci: Masalah Bratu, masalah Jenis-Bratu, kaedah penghuraian Adomian

asal, kaedah penghuraian Adomian terubahsuai, kaedah penembakan, kaedah

pembeza terhingga.

CONTENTS

DEDICATION

ACKNOWLEDGETvlENTS

ABSTRACT

ABSTRAK

CONTENTS

LIST OF TABLES

LIST OF FIGURES

LIST OF APPENDICES

CHAPTER TITLE

1. INTRODUCTION

1.1 Ivlotivation

1.2

1.3

1.4

1.5

1.7

Problem Statement

Objectives

Scope of the Research

1.4.1 The Accuracy of the Solution

1.4.2 The Stability of the rdethod

Outline of Thesis

Concluding Remarks

\'11

III

iv

\'

vi

Vll

xi

xii

xiii

PAGE

15

IG

16

17

I::::

2.

3.

\'i i i

LITERATURE REVIE\V 19

2.1

2.2

2.3

Introduction

Analytical Solution of the Bratu Problem

Analytical solution of the First Bratu-Type

Problem

19

2.4 Analytical solution of the Second Bratu-Type

Problem

2.5

2.6

2 '7 .1

2.8

Adomian Decomposition rvlethod :20

Shooting Ivlethod 28

Finite Difference IVlethod :31

Concluding Remarks

METHODOLOGY :3:3

3.1

3.2

3.3

Introduction

Procedures of the IVlethods

3.2.1 The Procedure of the Adomian Decomposition

Ivlethod

33

3.2.2 The Procedure of the Shooting lvlethod :3Ci

3.2.3 The Procedure of the Finite Difference l\Ict.hod :r,

Analysis of Adomian Decomposition J\Jethod

3.3.1 The Standard Adomian Decomposition ~ Ie! hod ·12

3.3.2 The l\Iodified Adomian Decomposition ~Ict hod ·11

3.3.3 Application of the Standard Aclomian Dccolllpu:;it inn

l\Iethod to Solving the Brntu Problem ·!G

ix

3.3.4 Application of the Standard Adomian

Decomposition IVlethod to Solving the

First Bratu-Type Problem 51

3.3.5 Application of the Standard Adomian

Decomposition Method to Solving the

Second Bratu-Type Problem 56

3.3.6 Application of the Ivlodified Adomian

Decomposition Method to Solving the

Bratu Problem 59

3.3.7 Application of the Modified Adomian

Decomposition Method to Solving the

First Bratu-Type Problem 63

3.3.5 Application of the lVIodified Adomian

Decomposition Ivlethod to Solving the

Second Bratu-Type Problem 66

3.4 Analysis of the Shooting Method 70

3.4.1 Application of the Shooting IVIethod to

Solving the Bratu Problem 74

3.4·4 Application of the Shooting Method to

Solving the First Bratu-Type Problem 75

3.4.3 Application of the Shooting Ivlethod to

Solving the Second Bratu-Type Problem 77

3.5 Analysis of the Finite Difference l'vlethod 79

3.5.1 Application of the Finite Difference :Method to

Solving the Bratu Problem 83

3.5.2 Application of the Finite Difference Ivlethod to

Solving the First Bratu-Type Problem 85

3.5.3 Application of the Finite Difference lvlethod to

Solving the Second Bratu-Type Problem 88

3.6 Concluding Remarks 91

4.

4.1

4.2

4.3

4.4

4.5

4.6

5.

NUMERICAL RESULTS AND DISCUSSION

Introduction

Numerical Results of the Bratu Problem

Numerical Results of the First Bratu-Type Problem

Numerical Results of the Second Bratu-Type Problem

The Accuracy of the Solutions

4.5.1 The Bratu problem

4.5.2 The First Bratu-Type problem

4.5.3 The Second Bratu-Type problem

The Stability of the lvlethods

SUMMARY AND CONCLUSIONS

5.1

5.2

5.3

Summary

Conrlusions

Suggestions for Further \Vorks

REFERENCES

x

93

9:3

93

100

102

104

104

105

106

107

108

108

109

110

111

xi

LIST OF TABLES

TABLE NO. TITLE PAGE

2.1 Corresponding values of e for various A ::; Ac 23

4.1 The upper solution of the shooting method

for the Bratu problem when A = 1 and A = 2 95

4.2 Comparispn between the exact solution (2.4) and the

four methods for A = 1 97

4.3 Comparison between the exact solution (2.4) and the

four methods for A = 2 99

4.4 The exact and numerical solutions of the

First Bratu-Type problem 101

4.5 The exact and numerical solutions of the

Second Bratu-Type problem 103

4.6 The accuracy of the Bratu problem 104

4.7 The accuracy of the First Bratu-Type problem 105

4.8 The accuracy of the Second Bratu-Type problem 106

XII

LIST OF FIGURES

FIGURE NO. TITLE PAGE

2.1 Dependence of solutions of equation (2.5) upon A 21

2.2 The exact solution of the Bratu problem 24

2.3 The exact solution of the First Bratu-Type problem 25

2.4 The exact solution of the Second Bratu-Type problem 26

4.1 The upper and lower solutions of the Bratu problem 94

4.2 The upper solutions of the shooting method

for the Bratu problem when A = 1 and A = 2 95

4.3 The Bratu problem for A = 1 97

4.4 The Bratu problem when A = 2 99

4.5 The First Bratu-Type problem 101

4.6 The Second Bratu-Type problem 103

xiii

LIST OF APPENDICES

APPENDIX TITLE PAGE

A The outline of the Maple algorithm for the standard Adomian

decomposition method in solving the Bratu and Bratu-Type

problem 114

B The outline of the Maple algorithm for the modified Adomian

decomposition method in solving the Bratu and Bratu-Type

problem 115

C The outline of the IVlaple algorithm for the shooting method

in solving the Bratu and Bratu-Type problem 116

D The outline of the Maple algorithm for the finite difference

method in solving the Bratu and Bratu-Type problem 118

E The relative errors of the four methods in

solving the Bratu and Bratu-Type problems 120

CHAPTER 1

INTRODUCTION

1.1 Motivation

\I\Then mathematical modeling is used to describe physical, biological or

chemical phenomena, one of the most common results of the modeling process is

a system of partial differential equations.

The Bratu problem is a partial differential equation which appears 1Il a

number of applications such as the steady state model of the solid fuel ignition

in thermal combustion theory and the Chandrasekhar model of the expansion of

the universe. The former model stimulates a thermal reaction process in a rigid

material, where the process depends on a balance between chemically generated

heat addition and heat transfer by conduction (Averick et al., 1992).

The classical Bratu problem can be described as follows:

o on D: {(x, Y) EO::; x ::; 1 , 0 ::; y ::; I} (1.1 )

with

11 - 0 on 3D,

where 6. is the Laplace operator and D is a bounded domain in TI2. According to

Jacobsen and Schmitt (2001), equation (1.1) arises in the study of the quasi linear

parabolic problem :

(1.2)

l/ 0, x E EJD,

which is also known as the solid fuel ignition model and is derived as a model for

the thermal reaction process in a combustible, nondeformable material of constant

density during the ignition period. Here A is known as the Fmnk-J(amendskii

1 parameter, l/ is a dimensionless temperature and - is the activation energy.

E

The derivation of equation (1.2) from general principles is accounted in the

comprehensive work by Frank and Kamenetskii in year 1955, who are interested

in what happens when combustible medium is placed in a vessel whose walls are

maintained at a fixed temperature. Intuitively, they expected that for a large

value of A, the reaction term will dominate and drive the temperature to infinity

(explosion) whereas for smaller /\ , the steady state might be possible.

Boyd (1986) developed a pseudo spectral method to generate approxiIllate

solutions to the classical two-dimensional planar Bratu problem

( 1.3)

on

{(x,y) E -1:S x:S 1, -1:S y:S I},

with lL = 0 on the boundary of the square. The basic idea is that the uJlkJlO\\"ll

solution u(x, y) can be completely represented as an infinite series of Sj)('ct ml

3

basis functions

N

u(x,y) L ak¢k(X, V)· (1.4 ) k=l

The basis functions ¢k(X, y) are chosen so that they obey the boundary condit.ions

and have the property that the more terms of the series are kept, the more

accurate the representation of the solution u(x, y) is. In other words, as N ~

00 the error diminishes to zero. For finite N, the series expansion in (1.4) is

substituted into (1.3) to produce the residual R. The residual function will depend

on the spatial variables x, y, the unknown coefficients ak and the parameter /\.

The goal of Boyd's pseudo spectral method is to find ak so that the residual

function R is zero at N collocation points. The collocation points are usually

chosen to be the roots of orthogonal polynomials that fall into the same family

as the basis functions ¢k(X, V). Boyd (1986) uses the Gegenbauer polynomials

to define the collocation points. The Gegenbauer polynomials are orthogonal on

the interval [-1,1] with respect to the weight nmction w(x) = (1 - X2)b where

b = - ~ corresponds to the Chebyshev polynomials and b = 1 is the choice Boyd 2

(1986) uses. The second order Gegenbauer polynomial is

3 2 -(5x - 1) 2' , -1 ~ x ~ 1.

Using a I-point collocation method at the point Xl = (1 _1_) and the choice v'5'v'5 of ¢1(X,y) = (1 - x2)(1 - y2), Boyd is able to obtain an approximation to the

value of /\c with a relative error of 8%. Note that this choice for 0] (x, y) satisfies

the boundary conditions since ¢(1, y) = ¢( -1, y) = 0(X, -1) = 0(X, 1) = O. The

solution produced by Boyd's pseudo spectral does not have the deficiency of being

wlable Lo converge to both solutions of the Bratu problem for /\ < /\c.

The Bratu problem in one-dimensional planar coordinates is also of tell

used as a benchmarking tool for numerical methods. The one-dimensional of this

problem is

ul/(X) + Aeu(x) O,O:S;x:S;l, (l.5)

with the boundary conditions

u(O) 0 and u(l) = o.

The nonlinear eigenvalue problem (1.5) has two known bifurcated exact solutions

for values of A < Ac, no solutions for A > Ac and a unique solution when /\ = /\,

where Ac = 3.513830719 is denoted as the critical value (Buckmire, 2003).

In Aregbesola (1996), the method of weighted residuals was used to solve

the Bratu problem (l.5). The idea is to approximate the solution with a polyno-

mial involving a set of parameters. The polynomial is of the form

N

V(x) = <I>o(x) + L A;<I>i(X), i=l

where <I>o(x) satisfies the given boundary conditions and each <I>i(X) satisfies the

homogenous form of the boundary conditions. The function V(x) is then used as

an approximation to the exact solution in the equation

L(U(x)) Q(x)

to give

R(x) L(U(x)) - Q(x).

where the function R(x) is the residual. The alIn is to make R(x) as small

as possible. One of the methods of minimizing R(x) is the collocation method

.5

where R(x) is set to zero at some points in the interval. The system of the

resulting nonlinear equations is then solved to determine the parameters Ai. The

polynomial V(x) is then considered as the approximate solution. The weighted

residual method provides accurate results and was found suitable for bifurcation

problems.

The availability of the exact solution of the Bratu problem (1.5) together

with universal applicability of the standard finite difference method, provides

an important application of the nonstandard finite difference method. Solving

a boundary value problem using the standard and nonstandard finite difFerence

methods involve replacing each of the derivatives by an appropriate difference-

quotient approximation. The interval [a, bJ is divided into N equal subintervals

where

a = Xo < Xl < X2 < ... < Xj < ... < XN = b.

1 For a uniform subintervals, the step size h is constant and h = N with Xi = a+ih

for i = 0,1,2, ... , N. The approximation of the second derivative by using the

centered-difference formula is

(1.6)

However in the nonstandard finite difference method, the denominator of (1.6),

11,2 is replaced by the denominator function ¢(h). Therefore, the nonstandard

fmite difference method for the second derivative is

" 11i+l - 211i + 11i-1 11 ~ ¢(h) (1. 7)

where the denominator function ¢(h) has the property that ¢(h) = h2 + 0(h2).

Buckmire (2003) has employed the standard finite difference method and the

6

nonstandard finite difference method for solving the Bratu problem (1.5). Using

the standard finite difference method, the discrete version of the Bratu problem

(1.5) is

U'i+l - 2U'i + Ui-l \ Ui - 0 h2 + /\e - , i = 1,2, ... ,N - 1. (1.8)

The nonstandard finite difference method for solving the Bratu problem (1.5) is

Ui+l - 2Ui + Ui-l A U

21n[cosh(h)] + e' = 0, i = 1,2, ... ,N - 1, (1.9)

where the denominator function, ¢(h) = 21n[cosh(h)] = h2 + O(h2). Thus, ill the

limit as h --+ 0, the standard finite difference method (1.8) and the nonstandard

finite difference method (1.9) will be identical.

Buckmire extended his research in the application of nonstandard finite

difference scheme to the cylindrical Bratu-Gelfand problem. The cylindrical

Bratu-Gelfand is a particular boundary value problem related to the classical

Bratu problem (1.1), with cylindrical radial operator. Jacobsen and Schmitt

(2002) considered the nature of solutions to a version of the classical Bratu prob-

lem (1.1) generalized to more complicated operators in more dimensions that

they called the Liouville-Bratu-Gelfand problem. The Liouville-Bratu-Gelfand

problem for the class of quasilinear elliptic equations is defined by

0, 0 < T < 1,

U > 0, (1.10)

1[.'(0) = u(l) 0,

7

where the inequalities a ::; 0, 1+ 1 > 0: and (3 + 1 > 0 hold. The Bratu-Gelfand

problem to be considered by Buckmire (2003) is the special case when 0: = 1,

(3 = 0, 1=1 :

1 ( ')' ,11 - ru + /\e r 0, 0 < r < 1,

u > 0,

u'(O) u(l) o.

(1.11)

The assumption has been made that u = u(r) in order to the other derivatives in

Laplacian can be ignored. In his previous works (Buckmire (1996) and (2003)),

he has shown that the usefulness in applying a particular nonstandard finite

difference scheme to boundary value problems in cylindrical coordinates that

contain the expression of r (~~). The expression r (~~) is then approximated by

the forward difference formula,

du Uk+l - Uk r-~rk----

dr rk+l - rk (1.12)

However, the following nonstandard finite difference scheme has been shown

(Buckmire, 2003) to be a superior method, especially for singular problems where

r --t 0:

(1.13)

Using the approximation in (1.12), the Bratu-Gelfand problem (1.11) will be

(1.14)

The nonstandard version finite difference scheme (1.13) for problem (1.11) will

be

o. (1.15)